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INTRODUCTION TO INSTABILITIES IN
NATURAL CIRCULATION SYSTEMS
P.K. Vijayan
Reactor Engineering Division,
Bhabha Atomic Research Centre, Mumbai, India
IAEA Course on Natural Circulation in Water-Cooled Nuclear Power
Plants, ICTP, Trieste, Italy, 25-29 June, 2007 (Lecture : T-6)
OUTLINE OF THE LECTURE T#6
- INTRODUCTION
- CLASSIFICATION OF INSTABILITIES
- STABILITY OF SINGLE-PHASE NC
- INSTABILITY DUE TO BOILING INCEPTION
- TWO-PHASE NC INSTABILITY
- CONCLUDING REMARKS
INTRODUCTION
Instabilities are common to both FC and NC systems
NCSs are more unstable. A regenerative feedback is inherent in the
mechanism causing NC
5.0
2.5
P - Pa
Any change in the driving force
will affect the flow which in turn
modifies the driving force leading
to a transient oscillatory state
even in cases which results in an
eventual steady state
0.0
-2.5
-5.0
500
1000
1500
2000
Time - s
Both single-phase and two-phase NCSs exhibit instability whereas
only two-phase FCSs are known to exhibit instability
Even two-phase NCSs are more unstable than two-phase FCSs
What is instability?
- Following a perturbation if
a thermal-hydraulic system
returns to its original steady
state, then the system is
considered to be stable
-If the system oscillates with
increasing amplitude or shift
to a new steady state, then it is
considered as unstable
Neutrally stable
Flow – kg/s
- If the system continues to
oscillate with the same
amplitude,
then
it
is
considered as neutrally stable
stable
unstable
Time - s
Limit Cycle Oscillations
3
0.2
nonlinearities
leading
to
limit
cycle oscillations
80 W
0.1
0.0
-0.1
-0.2
-3
1
Limit Cycle,0
Phase space (plot or
trajectory), orbit60 W
-1
Time series
System nonlinearities
P - mm of water column
Or
HHC
In bottom
all practical
systems
oscillation growth cannot continue
s the
horizontal
section
indefinitely. Instead, oscillation growth is terminated by2 system
-2
-1
0
1
2
3
- System temperature cannot be lower than the sink temperature
-2
- Void fraction can only vary between zero and unity
- Neutron flux cannot be negative
9000
12000
15000
18000
-3
0
3
DISADVANTAGES OF INSTABILITY
- Sustained flow oscillations may cause forced mechanical
vibration of components
- Premature CHF occurrence can be induced by flow
oscillations
- Induce undesirable secondary effects like power
oscillations in a BWR
- Instability can disturb control systems and pause
operational problems in nuclear reactors.
Classification of instabilities
Several kinds of instabilities are observed in NCSs excited by
different mechanisms
A fundamental cause of all instabilities is due to the existence
of competing multiple solutions so that the system is unable
to settle down to any of them permanently
However, differences exist in the transport mechanism,
oscillatory mode, phase shift, the nature of the unstable
threshold, and its prediction methods
Loop geometry and induced secondary phenomena also
affect the instability
Classification of instabilities – Contd.
Instabilities are classified according to various bases
- analysis method
- propagation method
- nature of the oscillations
- loop geometry,
- disturbances or perturbations
Classification of instabilities – Contd.
Analysis method (or Governing equations used)
(a) Pure (or fundamental) static instability
(b) Compound static instability
(c) Pure dynamic instability
(d) Compound dynamic instability
Pure static instability
The occurrence of multiple solutions and the instability threshold
itself can be predicted from the steady state equations governing
the process
Examples are Ledinegg instability and the instability induced by
the occurrence of CHF
Classification of instabilities – Contd.
Compound static instability
In some cases of multiple steady state solutions, the instability
threshold cannot be predicted from the steady state laws alone
or the predicted threshold is modified by other effects.
In this case, the cause of the instability lies in the steady state
laws, but feedback effects are important in predicting the
threshold
Typical examples are the instability due to flow pattern
transition, flashing and geysering
Classification of instabilities – Contd.
Pure Dynamic instability
For many NCSs neither the cause nor the threshold of instability
can be predicted from the steady state laws as inertia and feedback
effects are important
In this case, there are no multiple steady states, but the multiple
solutions appear during the transient
The full transient governing equations are required for explaining
the cause and predicting the threshold of instability
Typical example is the density wave oscillations commonly
observed in NCSs
Classification of instabilities – Contd.
Compound dynamic instability
In many oscillatory conditions, secondary phenomena gets
excited and it modifies significantly the characteristics and
the threshold of pure dynamic instability
In such cases even the prediction of the instability threshold
requires consideration of the secondary effect
Typical examples are
- neutronics responding to the void fluctuations
- primary fluid dynamics affecting the SG instability
Classification of instabilities – Contd.
Propagation method
This classification is restricted to only dynamic instabilities.
Dynamic instabilities involve propagation or transport of
disturbances. The disturbances can be transported by two kinds of
waves
- Pressure or acoustic waves
- Density waves (Stenning and Veziroglu)
In two-phase flow, both waves are present, however, their velocities
differ by one or two orders of magnitude allowing us to distinguish
between the two
DWI is the most commonly observed instability. The frequency of DWI
is of the order of 1 Hz in 2- flow
Classification of instabilities – Contd.
Based on the nature of oscillations - Flow excursions,
- Pressure drop oscillations,
- Power oscillations,
- Temperature excursions
Based on the periodicity
- Periodic oscillations
- chaotic oscillations
Based on the oscillatory mode
– Fundamental mode
- Higher harmonics
Based on the phase lag
Based on flow direction
- in-phase oscillations,
- out-of-phase oscillations
- dual oscillations
- Unidirectional oscillations
- Bi-directional oscillations
- Chaotic switching
Classification of instabilities – Contd.
Based on the loop geometry
- Open U-tube oscillations
C
C
F
- Pressure drop oscillations
0
Time
- Parallel channel oscillations
H
Based on the disturbances
Certain two-phase flow phenomena cause a major disturbance and
induce or modify the instability in NCSs
- Boiling inception
- Flashing and geysering
- Flow pattern transition
- Occurrence of CHF
Classification of instabilities – Contd.
Closure
The classification based on the analysis method is the most widely
accepted one and covers all observed instabilities
All other classifications addresses only a subset of the instabilities
It does not differentiate between NC and FC systems
Most instabilities observed in FC systems are observed in NCSs
However, certain instabilities associated with NCSs are not
observed in FCSs – Single-phase instability and the instability
associated with flow direction
Hence a specific discussion of instabilities for single-phase and twophase NCSs is considered useful
STABILITY OF SINGLE-PHASE NCSs
Single-phase NC instabilities can be characterised into three
different types
- Static instabilities associated with multiple steady states
- Dynamic instabilities
- Compound dynamic instabilities
Multiple steady states in the same flow direction
Pure Static instability
Multiple steady states with differing flow directions
Traditionally pure static instability is associated with multiple
steady states in the same flow direction
Theoretically certain single-phase NCSs show multiple steady states
in the same flow direction. So far no experimental confirmation
exists
STABILITY OF SINGLE-PHASE NCSs – Contd.
Multiple Steady states with differing flow direction
Unlike FC, certain NC systems can exhibit steady flow in both
clockwise and anticlockwise directions
coolant
Steady
Clockwise
flow (Q2=0)
Q2=0
Q1
Q2>Qc
Anticlockwise
flow with Q2>Qc
Q1
- Instability with oscillation growth will set in as Q2 nears Qc
- The instability will be terminated by flow reversal
- Subsequently even if we reduce power below Qc, instability will
not be observed. Also flow will continue in the reverse direction
- The instability cannot be predicted from steady state laws alone
Multiple steady states with differing flow directions
Instability associated with flow reversal in single phase NCS
1415
0.5
2100
1180
26.9
410
H
620
385
1x10
HHVC (ANTICLOCKWISE)
HHVC (CLOCKWISE)
23
0.0
stable
unstable
Grm
Grm
800
p - mm of water column
Lt=7.19m, Lt/D=267.29, p=0.316, b=0.25
C
Orientation : HHVC
10
-0.5
-1.0
13
Clockwise flow
Anticlockwise flow
3
10
9750
10000
10250
10500
Time - s
10750
11000
0
5
Stm
10
Stm
620
410
Even though the flow initiates in the anticlockwise direction, steady
flow was never obtained
Stability analysis showed that no steady flow exists with upward
flow in the cooler for this loop geometry
Upward flow in the cooler or downward flow in the heater can lead
to similar behaviour
Similar behaviour is observed in a NCS with throughflow
Multiple steady states with differing flow directions
Figure-of-eight loop with throughflow
Multiple steady states with feed in header 2 and bleed from header 3
compared with test data
Multiple steady states with differing flow directions
Instability near the flow reversal threshold: With small throughflow, the initial NC flow
direction is preserved. As the throughflow is large enough, it reverses the NC flow.
Near the flow reversal threshold an instability is observed
The amplitude of the oscillations
increase as the flow reversal threshold
approaches
F
F
F
F
W+F
W
Typical unstable behaviour near the flow reversal threshold.
The oscillation amplitude is
larger in the low flow branch
Experiments showed that
the flow reversal threshold
depends on the operating
procedure (hysteresis)
Parallel channel NC systems
Q
Outlet header
hl
downcomer
Q1 Q2 Q Q
3
n
W1 W W W
3
n
Qd=0
Wd
2
hm
Parallel vertical
inverted U-tubes
relevant to SGs
hs
Inlet header
Vertical parallel channel system relevant
to the RPV of BWRs, PWRs, etc.
cooler
ht
hm
Qt
hb
Qm
Qb
heater
Unequally heated parallel
horizontal channels having
unequal elevations relevant
to PHWRs
Existence
of
multiple
steady states was first
explored by Chato (1963)
Parallel Channel Systems
Unequally heated parallel channel NC systems exhibit an
instability associated with flow reversal due to the
existence of mutually competing driving forces
coolant
H
Q1
W1
W
Q2
W2
Downcomer
outlet header
W
inlet header
Vertical parallel channel system
There is a driving force between the
downcomer and each heated channel
promoting upward flow through the
heated channels
There is another driving force between
two unequally heated channels (due to
the difference in densities) favouring
downward flow in the low power
channel
The actual flow direction is decided
by the greater of the two driving forces
Parallel Channel Systems
For an unheated channel, upward flow is unstable and stable
downward flow can prevail.
Keeping Q2 constant, if we increase Q1 then flow reversal in
channel 1 occurs if Q1 > Qc (Chato (1963))
outlet header
coolant
H
Q1=0
W1
Keeping Q2 constant, if we reduce
Q1, then the upward flowing
channel 1 flow will reverse if
Q1<Qc (Linzer and Walter (2003))
W
Q2
W2
W
Down
comer
The flow reversal threshold is a
function of the power of channel 2.
Channel flow reversal is
inlet header
undesirable as it can lead to
(a) Vertical parallel channel system
instabilities in two-phase systems
Parallel Channel Systems
It appears that channel flow reversal can be avoided in a system of
vertical parallel channels if all the channels are equally powered
However, with vertical inverted U-tubes (as in SGs) and horizontal
channels as in PHWRs, mutually competing driving forces exist
even if all the channels are equally powered due to the differences
in elevation.
Single-phase NC with reverse flow in some of the longest U-tubes
are observed in several integral test loops
Steady state with one of the channels flowing in the reverse
direction was observed during thermosyphon tests in NAPS
1
A
Metastable
Regime
F
0.1
TOP HEADER
0.01
B
Ch-1
Ch-2
Ch-3
Decreasing power
Increasing power
C
D
-1.00
Rate of change of
power: 0.2%/s
DOWNCOMER
Power Ratio (Q1/Q2)
E
COOLANT OUTLET
COOLANT INLET
1E-3
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
Mass Flow Rate Ratio (W1/W2)
BOTTOM HEADER
Hysteresis curve for single-phase parallel channel NCS
computed with RELAP5/MOD3.2
 Initial steady state was achieved with equal power to Ch-1,2 & 3 and Ch-4 unheated.
 Ch- 1, 2 & 3 are with upflow and Ch-4 with downflow. Power in Ch-1 was decreased keeping other
channels’ power constant. The BLACK curve starts at A. After reaching a power ratio corresponds to B,
the flow in Ch-1 reverses from upflow (+ve) to downflow (-ve) and the curve jumps from state B to
state C.
 For the second case, initial steady state was achieved with equal power to Ch-2 & 3 and Ch-1 & 4
unheated. i.e. Ch-2 & 3 with upflow and Ch-1 & 4 with downflow.
 The BLUE curve starts at D. Power in Ch-1 was increased. After reaching a power ratio corresponds to
E, the flow in Ch-1 reverses from downflow (-ve) to upflow (+ve) and the curve jumps from state E to
state F.
Dynamic Instability in single-phase NCSs
Essentially the dynamic instability in single-phase systems is also
DWI although it was referred to as DWI only recently (Lahey, Jr)
The frequency of DWI in single phase NC (0.0015 – 0.005 Hz) is
significantly lower than that in two-phase systems (1-10Hz) due to
the low velocities in single-phase NC
Two types of dynamic instabilities are observed in single-phase NCSs
- Single channel system instabilities
- Parallel channel instabilities
Dynamic Instability in single-phase NCSs
System Instabilities
1415
Expansion
tank
C
cooler
Flow
305
800
Time
Keller (1966)
H
2200
3
St =7.0, Gr =1x1013
C
m
2
m
410
Heater
620
385
w
1
0
Flow
Time
-1
-2
H
-3
25
Welander
(1967)
50
75
t
100
410
620
All these oscillatory modes can be
observed in rectangular loops for
different ranges of power
Dynamic Instability in single-phase NCSs
Oscillation growth as a mechanism of single-phase instability
proposed by Welander (1967)
- Oscillation growth is the usual route to instability from steady state
- Compound static instability also shows oscillation growth,
but the instability is terminated by flow reversal
Parallel Channel Instability
Parallel channels also exhibit a dynamic instability mode in
single-phase conditions
The instability occurs due to the redistribution of flow
Both in-phase and out-of-phase oscillations are predicted
Two-phase NC instability
Pure static instability
Compound static instability
Dynamic instability
Compound Dynamic instability
Pure static instability
Flow excursion or excursive (Ledinegg) instability
Boiling Crisis
Ledinegg instability
Involves sudden change of flow rate to a lower value
The new flow rate may induce CHF
Occurrence of multiple steady states is the fundamental cause of
the instability
Ledinegg instability – Contd.
It occurs during the negative sloping region of the p – W
characteristic; d(p)/dW < 0 is the criterion for the Ledinegg
instability
Steam
Two-phase system characteristic
riser
Head
a
b
Feed
c
Driving pressure differential
heater
downcomer
Flow rate
Point ‘b’ satisfies this criterion and is therefore unstable
The instability can be avoided by inlet throttling in FCSs
Inlet throttling may not work as effectively as in FCSs due to the
reduction in flow caused by it
Boiling Crisis
Following the occurrence of critical heat flux, a regime of
transition boiling may be observed as in pool boiling
Heat flux
Transition
boiling
Film
boiling
Natural
convection
Nucleate
boiling
During transition boiling a film of
vapour prevents the liquid from
coming in direct contact with the
heating surface resulting in a steep
rise in temperature
Ts - Tsat
The film itself is not stable causing repetitive wetting and dewetting
of the heating surface resulting in an oscillatory surface temperature
The instability is characterized by sudden rise in wall temperature
followed by an almost simultaneous occurrence of flow oscillations
Compound Dynamic Instability
Two-phase NCSs also show an instability associated with flow
direction as in single-phase NC
All instabilities associated with boiling inception, flashing and
geysering, etc. can also be considered as part of two-phase NC
instability.
Flow pattern transition instability also belong to this category
Two-phase parallel channel systems exhibit
- Multiple steady states in the same flow direction
- An instability associated with flow reversal as in
single-phase flow
Flow Pattern Transition Instability
Two-phase systems exhibit different flow patterns with differences
in pressure drop characteristics which is the fundamental cause of
the instability
Bubbly-slug flow has a higher pressure drop compared to annular
flow
A system operating near the slug to annular transition boundary
is susceptible to this instability
Theoretical analysis of the phenomena is hampered by the
unavailability of validated flow pattern transition criteria and flow
pattern specific pressure drop correlations
The instability is found to be similar to Ledinegg instability, but
occurs at higher power
Dynamic instabilities in Two-phase NCSs
Fukuda-Kobori
1000
600
Single-phase
region
Type-II instability
80
2035 W
60
P
40
400
Pressure
200
1290
W
Upper
threshold
Type-I instability
0
0
100
4000
8000
20
0
12000
Time - s
Fig. 1a: Typical low power and high power
unstable zones for two-phase NC flow
Pressure - bar(g)
ΔP - Pa
800
Lower
threshold
Stable two-phase NC
Generally two unstable regions
are observed for DWI
The low power unstable region
is called the type I instability
The high power instability is
called the type II instability
The number of unstable or
stable zones depends on the
shape of the stability map
In view of the occurrence of islands of instability, the unstable zones
can be more than 2.
Boiling inception during stable single-phase flow
Stable single-phase flow can become unstable with the inception
of boiling
1.0
1 ba
r
15 b
The instability due to boiling
inception disappears at high
pressure
ar
Void fraction
0.8
70
bar
0.6
0.4
170
0.2
0.0
0.00
bar
220 b
0.05
ar
0.10
221.2 b
0.15
ar
0.20
Large variation of void fraction
with small changes in quality at
low pressures is the main cause
for the instability
Quality
Flashing and Geysering are also boiling inception instability
This is essentially the Type-I instability discussed by Fukuda
and Kobori.
Flashing Induced Instability
Occurs in NCSs with tall risers. The rising hot liquid experiences
static pressure decrease and may reach its saturation value leading
to flashing in low pressure systems
The increased buoyancy force increases the flow which in turn
reduces exit enthalpy and may even suppress flashing causing
reduction in buoyancy force and flow. Reduced flow leads to larger
exit enthalpy leading to the repetition of the process.
The necessary condition for flashing is that the fluid temperature
at the riser inlet is greater than the saturation temperature at the
exit
The instability is observed only in low pressure systems
Thermodynamic equilibrium conditions prevail during flashing
Geysering Instability
Generally observed in systems with tall risers. Geysering is
expected during subcooled boiling taking place at the exit of the
heater
As the bubbles move up the riser, bubble growth (due to static
pressure decrease) as well as condensation can take place.
Sudden condensation results in depressurisation causing the
liquid water to rush to the space vacated by the slug bubble
leading to a large increase in the flow rate reducing the driving
force
Geysering is a nonequilibrium phenomenon unlike flashing
Both Geysering and flashing instabilities are observed in low
pressure systems only
Dynamic Instability in Two-phase NCSs
Regenerative feedback and time delay are important for dynamic
instability
DWI is the most commonly observed dynamic instability in NCSs
and the mechanism causing the instability has already been discussed
earlier
A number of geometric and operating parameters affect the
instability in addition to certain boundary conditions
Riser height, orificing, length and diameter of source, sink and
connecting pipes are the important geometric parameters
Pressure, inlet subcooling, power and its distribution are the
important operating parameters
Boundary conditions of interest are wall heat transfer coefficient,
heat storage in walls, wall heat losses, constant pressure drop, etc.
Compound Dynamic Instability in two-phase
NCSs
If only one instability mechanism is at work, it is said to be fundamental
or pure instability. Instability is compound if more than one
mechanisms interact in the process and cannot be studied separately
-Thermal oscillations
-Parallel channel instability
-Pressure drop oscillations
-BWR instability
-SG instability
Thermal Oscillations
The variable heat transfer coefficient leads to a variable thermal
response of the heated wall that gets coupled with the DWO
Thermal oscillations are a regular feature of post dry out heat transfer
heat transfer in steam-water mixtures at high pressures
The steep variation in heat transfer coefficient gets coupled with DWO
The dryout or CHF point shift upstream of or downstream during
thermal oscillations
The large variation in heat transfer coefficient results in significant
variation in heat transfer rate even if the wall heat generation is constant
Parallel Channel Instability
Interaction of parallel channels with DWO can give rise to interesting
behaviours as in single-phase NCS
-In-phase oscillations (system instability)
- Out-of-phase oscillations
-Dual oscillations (overlapping region of IPO and OPO)
In-phase oscillation is a system characteristic and parallel channels
may not play a role in this
Out-of-phase oscillation is characteristic of parallel channels. The
phase shift depends on the number of channels
- 2 channels : 1800 ; 3 channels : 1200 ; n-channels :2/n
However, in a system of n-channels, all parallel channels need not
take part in the instability. Depending on the number of channels
taking part the phase shift can be anywhere between 2/n to 
Mechanism of PCI is similar in single-phase and two-phase systems
Pressure drop oscillations
This is associated with operation in the negative sloping region of
p-w curve.
Caused by the interaction of a compressible volume at the inlet of the
heated section with the pump characteristics
Usually observed in FCSs. DWO oscillation occurs at flow rates
lower than the flow rate at which PDO is observed
The frequency of PDO is much smaller than DWO.
Very long test sections may have sufficient internal compressibility
to cause PDO
Like Ledinegg instability there is a danger of CHF occurrence
during PDO
Inlet throttling can stabilize PDO as in the case of Ledinegg
instability
Instability in NBWRs
The flow velocity in NBWRs is usually much smaller than
FC BWRs
The presence of tall risers causes the oscillation frequency to be
much smaller
The only NCR whose stability has been extensively studied is the
Dodewaard reactor.
The negative void reactivity stabilizes type I instability. But it may
stabilize or destabilize type II instability
Pump trip transients in FC BWRs lead to type II instability
CONCLUDING REMARKS
Various bases used for classification of instabilities have been
reviewed. The most widely accepted classification is based on the
method of analysis used in identifying the stability threshold.
While classifying NC instabilities, it was convenient to consider the
instabilities associated with single-phase and two-phase NC
separately
Natural circulation systems are more unstable due to the
regenerative feedback inherent in the mechanism causing the flow.
Besides the instability in single-phase systems, natural circulation
loops also exhibit an instability associated with flow reversal in
contrast to forced circulation systems.
Thank you
Classification of instabilities – Contd.
Density wave instability
DWI is most commonly observed in NCSs. The frequency of DWI is
of the order of 1 Hz in 2- flow
Due to the importance of void fraction and its effect on flow, the
instability is often referred to as ‘flow-void feedback instability’ in
two-phase NCSs
Since transportation time delays are crucial to the instability, it is
also known as ‘time delay oscillations’
In single-phase, near critical and supercritical fluids, the instability
is also known as ‘thermally induced oscillations’
Instabilities Associated with Boiling Inception
Boiling inception modifies single-phase instability
Boiling inception can induce instabilities in a stable single-phase
NCS
Occurrence of single-phase conditions during part of the
oscillation cycle is a characteristic feature of this instability
Effect of Boiling inception on single-phase instability
With power increase boiling inception occurs during the low flow part of the
oscillation cycle. Instability continues with flow switching between singlephase and two-phase regimes.
Several flow regimes are observed as shown. The change in power required for
attaining the condition with two-phase flow for the entire oscillation cycle can
be very significant
0.2
0.2
Orientation : HHHC
Plot for 21950 seconds after neglecting initial transients
Orientation : HHHC
Plotted from a time series of 7900 seconds after neglecting initial transients
0.1
Flow rate - kg/s
Flow rate - kg/s
0.1
0.0
0.0
-0.1
-0.1
-0.2
-3
-2
-1
0
1
2
-0.2
-3
3
-2
P - mm of water column
(a) Single-phase instability
-1
0
1
P - mm of water column
2
3
(b) Instability with sporadic boiling
0.3
0.2
Orientation : HHHC
Plotted from a time series of 10000 seconds after neglecting initial transients
Orientation : HHHC
Plotted from a time series of 19500 seconds
after neglecting initial transients
0.2
Flow rate - kg/s
Flow rate - kg/s
0.1
0.0
-0.1
0.1
0.0
-0.1
-0.2
-0.2
-3
-2
-1
0
1
P - mm of water column
2
c) Instability with boiling once in
every cycle
3
-0.3
-3
-2
-1
0
1
P - mm of water column
2
3
(d) Instability with boiling twice
in every cycle